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Theorem icomnfordt 20163
Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
icomnfordt  |-  ( -oo [,) A )  e.  (ordTop `  <_  )

Proof of Theorem icomnfordt
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2429 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2429 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 20160 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 20153 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2514 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 19917 . . . . . . 7  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  <-> 
( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top )
86, 7mpbir 212 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 19912 . . . . . 6  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  ->  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
) )
108, 9ax-mp 5 . . . . 5  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
1110, 4sseqtr4i 3503 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3635 . . . . 5  |-  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) 
C_  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
13 ssun2 3636 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2429 . . . . . . . 8  |-  ( -oo [,) A )  =  ( -oo [,) A )
15 oveq2 6313 . . . . . . . . . 10  |-  ( x  =  A  ->  ( -oo [,) x )  =  ( -oo [,) A
) )
1615eqeq2d 2443 . . . . . . . . 9  |-  ( x  =  A  ->  (
( -oo [,) A )  =  ( -oo [,) x )  <->  ( -oo [,) A )  =  ( -oo [,) A ) ) )
1716rspcev 3188 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( -oo [,) A )  =  ( -oo [,) A
) )  ->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
1814, 17mpan2 675 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
19 eqid 2429 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( -oo [,) x ) )  =  ( x  e.  RR*  |->  ( -oo [,) x ) )
20 ovex 6333 . . . . . . . 8  |-  ( -oo [,) x )  e.  _V
2119, 20elrnmpti 5105 . . . . . . 7  |-  ( ( -oo [,) A )  e.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) )  <->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
2218, 21sylibr 215 . . . . . 6  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
2313, 22sseldi 3468 . . . . 5  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3468 . . . 4  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3468 . . 3  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 467 . 2  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 11641 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 11645 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5751 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6467 . . 3  |-  ( -.  ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A
)  =  (/) )
31 0opn 19865 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2525 . 2  |-  ( -.  ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 167 1  |-  ( -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783    u. cun 3440    C_ wss 3442   (/)c0 3767   ~Pcpw 3985    |-> cmpt 4484    X. cxp 4852   ran crn 4855   ` cfv 5601  (class class class)co 6305   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673    < clt 9674    <_ cle 9675   (,)cioo 11635   (,]cioc 11636   [,)cico 11637   topGenctg 15295  ordTopcordt 15356   Topctop 19848   TopBasesctb 19851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-topgen 15301  df-ordt 15358  df-ps 16397  df-tsr 16398  df-top 19852  df-bases 19853  df-topon 19854
This theorem is referenced by: (None)
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